Just about everyone. You said yourself that you were ostracized because of this idea.

> > You think that they are able to grasp the idea of subtraction as the> inverse of addition if and only if it is not written using the letters> of the alphabet?

They don't understand what "inverse" means.

> > Can't at least many of them understand that 10 - 3 = 7 is true when> and only when and 10 = 7 + 3 is true?

They don't understand what "true" means. They might agree with you, but they don't understand it.

> > Can't at least many of them understand that we can replace these> numbers in such examples with letters of the alphabet a,b,c to express> the idea being true for all numbers, that a - b = c is true when and> only when a = c + b is true?

They certainly don't understand what "true for all numbers" means. They might agree with you but they don't understand it.

> > And if so, can't they all see the visual attribute of the written> expression of the definition of subtraction as the inverse of> addition, that a number or variable looks like it moves back and forth> the equality symbol and the sign of the number changes as it does?

No, they cannot understand this. They might agree with you, but they don't understand it.

A key element of understanding is appreciating what it means, not nodding your head and agreeing with you.

You are confusing the wrapping paper with the package inside. If mathematics worked like this, then we wouldn't have this forum. The vast majority of students wouldn't fail to get algebra. Your notions are understandable and I don't expect any of us to entirely stop "reaching", but when these notions make it into standards, it is bad. It is a signal of just how botched the task of teaching mathematics has become. The real bad guy isn't this "pretending", it is conceptualization, which I think stems from the same lack of "thinking the pedagogy through". Nothing has done more damage to actual mathematical understanding than the notion of conceptualization. Ironic, isn't it.

Lately, I am on a tear. I have not had the urge to pretend much. I owe that to the realization last year that the curriculum's biggest fault was its ignorance of numbers. After going through the trouble of learning the times table, it digressed to pictures (which I have criticized here previously). I chose instead to go on to division, factoring, prime numbers, perfect squares, LCM and GCD. So while the curriculum teaches a trick for estimating (that the students cannot appreciate), I teach about magnitude and precision, in terms that a fourth grader can understand and sense (see attachment).

The result is that the teachers say to me that Michael is so advanced in math. What I find they mean is that he understands numbers and can do arithmetic well. I want say to them "Have you not thought about teaching the students numbers and arithmetic?", but I don't. Some of them probably have thought about it, but they don't have a choice (state standards), and it has been so long since teachers did teach about numbers and arithmetic, most probably don't remember how anymore.